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pg 3: @10:34 Lowering factorial power; Raising factorial power; connection between raising and lowering; all polynomials @;
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Polynomials and Sequence Spaces - Wild Linear Algebra - NJ Wildberger
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- 1 CONTENT SUMMARY: pg 1: @
- 2 polynomials and sequence spaces; remark about expressions versus objects @ ;
- 3 pg 2: @ Some polynomials and associated sequences; Ordinary powers; Factorial powers D. Knuth;
- 4 pg 3: @10:34 Lowering factorial power; Raising factorial power; connection between raising and lowering; all polynomials @;
- 5 pg 4: @ Why we want these raising and lowering factorial powers; general sequences; On-line encyclopedia of integer sequences N.Sloane; 'square pyramidal numbers'; Table of forward differences;
- 6 pg 5: @19:23 Forward and backward differences; forward/backward difference operators on polynomials; examples: operator on 1 @;
- 7 pg 6: @ Forward and backward differences on a sequence; difference below/above convention;
- 8 pg 7: @27:21 Forward and backward Differences of lowering powers; calculus reference @;
- 9 pg 8: @31:27 Forward and backward Differences of raising powers; operators act like derivative @ ; n equals 0 raising and lowering defined;
- 10 pg 9: @ Introduction of some new basis; standard/power basis, lowering power basis, raising power basis; proven to be bases;
- 11 pg 10: @ WLA22_pg10_Theorem Newton; proof;
- 12 pg 10b: @44:40 Lesson: it helps to start at n=0; example square pyramidal numbers;an important formula @;
- 13 pg 11: @50:00 formula of Archimedes; taking forward distances compared to summation @
- 14 pg 12: @ a simpler formula; example: sum of cubes;
- 15 pg 13: @ exercises 22.1-4;
- 16 pg 14: @59:06 exercise 22.5; find the next term; closing remarks @;
- 17 Introduction
- 18 Some polynomials and associated sequences
- 19 Lowering factorial powers
- 20 Forward and backward differences
- 21 Differences of lowering and raising powers are easy to compute!
- 22 Factorial power bases
- 23 A theorem of Newton
- 24 A formula of Archimedes
- 25 A formula for sum of cubes
- 26 Exercises 22.1-4;
